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Thermal instability of rotating nanofluid layer
International Journal of Engineering Science 49 (2011) 1171–1184
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Thermal instability of rotating nanofluid layer Dhananjay Yadav ⇑, G.S. Agrawal, R. Bhargava Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India
a r t i c l e
i n f o
Article history: Received 18 November 2010 Received in revised form 2 July 2011 Accepted 13 July 2011 Available online 23 August 2011 Keywords: Nanofluid Brownian motion Thermal instability Galerkin method Critical Rayleigh number Thermophoresis
a b s t r a c t In the present paper we have considered thermal instability of rotating nanofluids heated from below. Linear stability analysis has been made to investigate analytically the effect of rotation. The more important effect of Brownian motion and thermophoresis has been included in the model of nanofluid. Galerkin method is used to obtain the analytical expression for both non-oscillatory and oscillatory cases, when boundaries surfaces are free–free. The influence of various nanofluids parameters and rotation on the onset of convection has been analysed. It has been shown that the rotation has a stabilizing effect depending upon the values of various nanofluid parameters. The critical Rayleigh number for the onset of instability is determined numerically and results are depicted graphically. The necessary and sufficient conditions for the existence of over stability are also obtained. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Nanofluid is the suspension of nanoparticles in a base fluid, which was first utilized by Choi (1995). Nanoparticles used in nanofluid preparation usually have diameters below 100 nm. Due to their small size, nanoparticles fluidize easily inside the base fluid, and as a consequence, blockage of channels and erosion in channel walls are no longer a problem. Nanoparticles materials include oxide ceramics (Al2O3, CuO), metal carbides (SiC), nitrides (AlN, SiN), metals (Al, Cu) etc. As mentioned in the literature, base fluid include water, ethylene or tri-ethylene-glycols and other coolants, oil and other lubricants, biofluids, polymer solutions, other common fluids. In order to improve the stability of nanoparticles inside the base fluid, some additives are added to the mixture in small amounts. Buongiorno (2006) studied the convective transport in nanofluids and observed that the absolute velocity of the nanoparticles can be viewed as the sum of the base fluid velocity and a relative velocity. He also discussed the effect of seven slip mechanisms: inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage, and gravity setting. He concluded that in the absence of turbulent eddies, Brownian diffusion and thermophoresis dominates the other slip mechanisms. The Rayleigh–Bénard problem for a regular fluid with rotation or without rotation was first discussed by Chandrasekhar (1961). Stability of a micro polar fluid layer heated from below has been studied by Ahmadi (1976). Qin and Kaloni (1992) have considered a thermal instability problem in rotating micro polar fluids. They obtained that for low values of Taylor number rotation has a stabilizing effect. Thermal instability problem for nanofluid without rotation was studied by Tzou (2008a, 2008b) and Dhananjay, Agarwal, and Bhargava (2011). The onset of convection in a horizontal nanofluid layer of finite depth was studied by Nield and Kuznetsov (2010). Alloui, Vasseur, and Reggio (2010) studied the natural convection of nanofluids in a shallow cavity heated from below. They observed that the presence of nanoparticles in a fluid is found to reduce the strength of flow field, this behavior being more ⇑ Corresponding author. Tel.: +91 9286685908. E-mail addresses: [email protected] (D. Yadav), [email protected] (G.S. Agrawal), [email protected] (R. Bhargava). 0020-7225/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2011.07.002
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Nomenclature a wave number c specific heat d diameter of nanoparticles D distance (m) DBT diffusion coefficient (m2/s) g gravitation (m/s2) hp = cprT enthalpy (j/kg) of nanoparticles j mass flux (kg/m2s) k thermal conductivity (w/mK) Boltzmann constant (j/K) kB p pressure (Pa) P pressure t time (s) ui velocity (m/s), i = 1, 2, 3 Ui velocity,i = 1, 2, 3 xi space (m), i = 1, 2, 3 coordinate, i = 1, 2, 3 Xi Pr Prantdl number Ra Rayleigh number Greek symbols / volume fraction of nanoparticles a thermal diffusivity (m2/s) b thermal expansion coefficients (1/K) q mass density (kg/m3) d Kronecker delta Z x3-component of vorticity. s time
pronounced at low Rayleigh number. Also, the temperatures on the solid boundaries are reduced (enhanced) by the presence of the nanoparticles when the strength of convection is high (low). For completeness a substantially different treatment of Rayleigh- Bénard problem for nanofluids was given by Kim, Kang, and Choi (2004). In this paper the quantities namely the thermal expansion coefficients, the thermal diffusivity, and kinematic viscosity that appear in the definition of Rayleigh number were modified with respect to the nanofluid. The effect of rotation on the Rayleigh–Bénard convection in nanofluids is important in certain chemical engineering and biochemical engineering. Rayleigh–Bénard convection in rotating nanofluids about a vertical axis combines the element of thermal buoyancy and rotation induced Coriolis and centrifugal forces. Due to the Coriolis force on the Rayleigh–Be‘nard convection another parameter namely Taylor number is introduced in this problem. Taylor number is a non-dimensional number which is a measure of rotation rate. It is apparent that Rayleigh–Bénard convection in rotating nanofluids will play an important role in many physical phenomenon concerning with geophysics, astrophysics and oceanography. In the present paper we have studied the thermal instability of rotating nanofluid layer with free boundaries. We have obtained the critical Rayleigh number as well as critical wave number by using Galerkin-type weighted residuals method. Stability is discussed analytically as well as numerically. It has been observed that the temperature gradient and rotation have stabilizing effect, while volumetric fraction of nanoparticles and the density ratio of nanoparticles vis-a-vis base fluid have destabilizing effect on the system. We have also discussed the case of over stability and compared our results with that of Chandrasekhar (1961). 2. Problem formulation Consider an infinite horizontal layer of incompressible nanofluid which is kept rotating about vertical axis at a constant angular velocity X = (0, 0, X) and heated from below. Let us consider the Cartesian co-ordinate system x1, x2, x3 in which x3 is taken at right angle to the boundaries. The nanofluid is confined between two parallel plates x3 = 0 and x3 = L, where temperature and volumetric fraction of nanoparticles are kept constants: T = T0 and / = /0 at x3 = 0 and T = T1, / = /1 at x3 = L (shown in the Fig. 1). We assume that the both boundaries surfaces are free. We have taken the thermo-physical properties of nanofluids (viscosity, density, thermal conductivity, and specific heat) as constants for the analytical formulation but these quantities are not constant and strongly depend on the volume fraction of nanoparticles. Under the Boussinesq approximation (Rajagopal, Saccomandi, & Vergori, 2009), the continuity and momentum equations are
D. Yadav et al. / International Journal of Engineering Science 49 (2011) 1171–1184
1173
x3 axis Ω = (0,0, Ω)
x3 = L, φ = φ1
T1
- - - - - - - - - - - - - - - - - - - - - - - - - - - --- g = (0,0, − g ) - Incompressible
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- -
rotating nanofluid - - - - - - -- - - x2 axis
-- - - - - - - - - -- - - - - - - - - - - - -
T0
O
x1 axis
x3 = 0, φ = φ0
Fig. 1. Sketch of the problem geometry and coordinates.
r:V ¼ 0;
@ q0 þ ðV:rÞ V ¼ rp þ lr2 V qgk þ 2q0 V X; @t
ð1Þ ð2Þ
where
q0 /0 ¼ /qp þ ð1 /Þqf0 ; q ¼ /qp þ ð1 /Þqf ¼ /qp þ ð1 /Þqf0 ½1 bðT T 0 Þ;
ð3Þ
q0 and qf0 being the nanofluid density and base fluid density at reference temperature T0 respectively.
Taking the density of the nanofluid as that of the base fluid, as adopted by Tzou (2008a, 2008b), the specific weight (qg) in Eq. (2) thus becomes
qg ffi ½/qp þ ð1 /Þ½q0 f1 bðT T 0 Þgg:
ð4Þ
Due to Brownian motion and thermoporesis, the mass flux (jp) of the nanoparticles in base fluid is given by
jp ¼ qp DB r/ qp
DT rT; T
ð5Þ
where DB is the Brownian diffusion coefficient, given by Einstein-Stokes equation, and DT is the thermoporetic diffusion coefficient of the nanoparticles given as:
DB ¼
kB T ; 3pldp
DT ¼
l q
0:26k /: 2k þ kp
ð6Þ
In the linear theory, temperature change in the nano fluid is less compared to the reference temperature (T0) (Tzou, 2008a, 2008b). Therefore, nanofluid temperature T in the Eq. (5) has been replaced by T0. The continuity equation for the nanoparticles is given by:
@ 1 DT þ ðV:rÞ / ¼ r:jp ¼ DB rðr/Þ þ rðrTÞ: @t qp T0
ð7Þ
The energy equation is:
q0 c
@ þ ðV:rÞ T ¼ r:q þ hp r:jp ; @t
ð8Þ
where hpr.jp is the addition flow work in energy equation due to Brownian motion and thermophoresis of nanoparticles relative to the flow velocities (Buongiorno, 2006) and
q ¼ krT þ hp jp ) r:q ¼ kr2 T þ r:ðhp jp Þ ¼ kr2 T þ hp r:jp þ jp :rhp ¼ kr2 T þ hp r:jp þ jp :ðcp rTÞ; where rhp is equal to cprT. Substituting the expression for r.q and jp, the energy Eq. (8) becomes as:
q0 c
@ DT ðrTÞ2 ; þ ðV:rÞ T ¼ krðrTÞ þ qp cp DB ðr/ÞðrTÞ þ @t T0
and vorticity is given by:
ð9Þ
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n ¼ r V:
ð10Þ
Eqs. (1), (2), (7), (9) and (10) gives seven equations for seven unknown: three velocity components (u1, u2, u3), pressure (p), volumetric fraction of nanoparticles (/), temperature (T) and x3- component of vorticity. Introducing the following non dimensional parameters
Xi ¼
xi ; L
s¼
t L2
a; P ¼
p L2 ; ðq0 a2 Þ
/¼
/ /1 ; /0 /1
h¼
T T1 ; T0 T1
Ui ¼
ui L; ðaÞ
Z¼
n L2
a;
ð11Þ
the non dimensional form of Eqs. (1), (2), (7), (9) and (10) become as:
@U 1 @U 2 @U 3 þ þ ¼ 0; @X 1 @X 2 @X 3 pffiffiffiffiffi @U 1 @U 1 @U 1 @U 1 @P þ U1 þ U2 þ U3 ¼ þ P r r2 U 1 þ P r T a U 2 ; @X 1 @s @X 1 @X 2 @X 3 pffiffiffiffiffi @U 2 @U 2 @U 2 @U 2 @P þ U1 þ U2 þ U3 ¼ þ P r r2 U 2 P r T a U 1 ; @X 2 @s @X 1 @X 2 @X 3
ð12Þ ð13Þ ð14Þ
@U 3 @U 3 @U 3 @U 3 @P þ U1 þ U2 þ U3 ¼ þ Pr r2 U 3 H½f1 þ bðT 1 T 0 Þgð/1 1Þ þ Rq /1 Ra Pr ð/1 1Þh @X 3 @s @X 1 @X 2 @X 3 H½Rq 1 þ bðT 1 T 0 Þð/0 /1 Þ/ Ra Pr ð/0 /1 Þ/h; with H ¼
ð15Þ
Ra P r bðT 0 T 1 Þ
@/ @/ @/ @/ þ U1 þ U2 þ U3 ¼ N BT r2 / þ NTT r2 h; @s @X 1 @X 2 @X 3
ð16Þ
@h @h @h @h 1 @h @/ @h @/ @h @/ þ þ þ U1 þ U2 þ U3 ¼ r2 h þ @s @X 1 @X 2 @X 3 Le @X 1 @X 1 @X 2 @X 2 @X 3 @X 3 " 2 2 2 # RN @h @h @h ; þ þ þ @X 1 @X 2 @X 3 Le
ð17Þ
and
Z¼
@U 2 @U 1 : @X 1 @X 2
ð18Þ
In the terms of the non-dimensional form, the boundary condition becomes as:
h ¼ 1 and / ¼ 1 at X 3 ¼ 0
ð19Þ
h ¼ 0 and / ¼ 0 at X 3 ¼ 1: The nanofluid is thus characterized by eight parameters:
Ra ¼ gL3 b Ta ¼ RN ¼
4X2
m2
ðT 0 T 1 Þ
am
ðRayleigh numberÞ;
L4 ðTaylor numberÞ;
NTT ; NBT
NTT ¼
Pr ¼
Le ¼
k
qp cp DB ð/0 /1 Þ
ðLewis numberÞ;
t DB ðPrantdl numberÞ; NBT ¼ ; a a
Rq ¼
qp ; q0 ð20Þ
DT ðT 0 T 1 Þ : T 0 a ð/0 /1 Þ
2.1. Primary flow Assuming a time-independent solution of Eqs. (12)–(18), the primary flow is supposed to be stationary i.e. U i ¼ 0 for are varying in X3 direction only. From the Eqs. (15)–(17), i = 1,2,3, whereas temperature ð hÞ and nanoparticles fraction ð/Þ the equations governing the primary flow are: 2 d /
þ RN 2
2 d h
¼ 0; 2 dX 3 2 1 d/ dh RN dh þ þ ¼ 0; 2 dX 3 Le dX 3 dX 3 Le dX 3 dX 3 2 d h
ð21Þ ð22Þ
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1175
@P þ Ra Pr ð/ / Þh/: ¼ H½f1 þ bðT 1 T 0 Þgð/1 1Þ þ Rq /1 þ Ra P r ð/1 1Þh þ H½Rq 1 þ bðT 1 T 0 Þð/0 /1 Þ/ 0 1 @X 3 ð23Þ
3 Þ are The boundary condition for hðX 3 Þ and /ðX
hð0Þ ¼ 1; /ð0Þ ¼ 1 and hð1Þ ¼ 0; /ð1Þ ¼ 0:
ð24Þ
Integrating Eq. (21) with respect to X3 we have
¼ RN h þ c1 X 3 þ c2 ; /
ð25Þ
where c1 and c2 are integration constants. Substituting Eq. (25) into the Eq. (22), we obtain 2 d h 2 dX 3
2
þ
c1 d h ¼ 0: Le dX 23
ð26Þ
From Eqs. (24)–(26) we have,
" hðX 3 Þ ¼
RN exp
n
ð1þRN ÞX 3 Le
o
þ exp
# 1þRN f1 ð1 þ RN ÞX 3 g Le
exp
þð1 þ RN ÞðX 3 1Þ
1 1 þ RN 1 ; Le
1 1 þ RN 3 Þ ¼ 1 exp ð1 þ RN Þð1 X 3 Þ 1 exp : /ðX Le Le
ð27Þ
ð28Þ
According to Buongiorno (2006), for most of the nanofluids RN 1 10,Le 104 105 and e = (1 + RN)/Le 105 104. in power series of e, we have, Expanding h and /
hðX 3 Þ ¼ 1 X 3 þ RN X 3 ðX 3 1Þ e þ ; 2 X ð1 X3Þ 3 3Þ ¼ 1 X3 þ /ðX e þ ; 2
ð29Þ ð30Þ
Hence, we have with e 104, as compared to (1 X3) 100, the zeroth order terms are dominant in both h and /. hðX 3 Þ ¼ /ðX 3 Þ 1 X 3 , which display linear distributions in X3. 2.2. Equations after perturbation For small disturbances onto the primary flow, we assume that
U i ¼ U i þ U 0i ;
q ¼ q þ q0 ; P ¼ P þ P 0 ; h ¼ h þ h0 ; Z ¼ Z þ Z0 and / ¼ / þ /0 ;
ð31Þ
3Þ 1 X3. with hðX 3 Þ ¼ /ðX Substituting Eq. (31) into Eqs. (12)–(18) and neglecting the product of prime quantities, we have
@U 01 @U 02 @U 03 þ þ ¼ 0; @X 1 @X 2 @X 3 pffiffiffiffiffi @U 01 @P0 ¼ þ Pr r2 U 01 þ Pr T a U 02 ; @s @X 1 pffiffiffiffiffi @U 02 @P0 ¼ þ P r r2 U 02 Pr T a U 01 ; @s @X 21 @U 03 @P0 0 Þ; ¼ þ Pr r2 U 03 Ra Pr ð/1 1Þh0 ð/0 /1 Þ½HfRq 1 þ bðT 1 T 0 Þg/0 þ Ra P r ðh/0 þ /h @s @X 3 @/0 U 03 ¼ N BT r2 /0 þ NTT r2 h0 ; @s 0 0 @h0 1 @/ @h 2RN @h0 þ ; U 03 ¼ r2 h0 Le @X 3 @s @X 3 Le @X 3 0 0 @U 2 @U 1 : Z0 ¼ @X 1 @X 2 The above system will be used in further analysis.
ð32Þ ð33Þ ð34Þ ð35Þ ð36Þ ð37Þ ð38Þ
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3. Normal mode analysis Assuming the perturbation quantities of the form as:
0 0 0 0 0 0 0 U 1 ; U 2 ; U 3 ; P ; h ; / ; Z ¼ ½U 1 ; U 2 ; U 3 ; P; h; /; ZðX 3 Þ exp½ikX 1 X 1 þ ikX2 X 2 þ rs;
ð39Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 are the wave number along the X1 and X2 directions respectively, and a ¼ kX 1 þ kX 2 is the resultant wave
where kX 1 ; kX 2 number. Following the normal mode analysis, the linearized equations in dimensionless form are as follows:
U 1 ikX 1 þ U 2 ikX 2 þ DU 3 ¼ 0;
ð40Þ
pffiffiffiffiffi þ P r ðD a ÞU 1 þ Pr T a U 2 ; pffiffiffiffiffi þ P r ðD2 a2 ÞU 2 Pr T a U 1 ; 2
2
rU 1 ¼ PikX1 rU 2 ¼ PikX2 rU 3 ¼ DP þ Pr ðD2 a2 ÞU 3 Ra Pr ð/1 1Þh ð/0 /1 Þ½HfRq 1 þ bðT 1 T 0 Þg/ þ Ra Pr ðh/ þ /hÞ;
ð42Þ ð43Þ
r/ U 3 ¼ NBT ðD2 a2 Þ/ þ NTT ðD2 a2 Þh; rh U 3 ¼ ðD2 a2 Þh
ð41Þ
ð44Þ
1 2RN ðDh þ D/Þ Dh; Le Le
ð45Þ
Z ¼ ikX 1 U 2 ikX 2 U 1 ;
ð46Þ
where D dXd 3 . From Eqs. (40)–(42) and (46) the expression for P can be obtained as:
pffiffiffiffiffi Pr ðD2 a2 ÞDU 3 rDU 3 Pr T a Z P¼ : a2
ð47Þ
Substituting Eq. (47) in Eq. (43), we have:
pffiffiffiffiffi T a DZ þ a2 Ra Pr ð/1 1Þh þ a2 ð/0 /1 Þ þ /hÞ ¼ 0: ½HfRq 1 þ bðT 1 T 0 Þg/ þ Ra Pr ðh/
ðD2 a2 Þ½Pr ðD2 a2 Þ rU 3 P r
ð48Þ
On elimination of P from Eqs. (40)–(42) and (46) we get:
½P r ðD2 a2 Þ rZ þ Pr
pffiffiffiffiffi T a DU 3 ¼ 0:
ð49Þ
From Eqs. (48) and (49), we have:
ðD2 a2 Þ½Pr ðD2 a2 Þ r2 U 3 þ P2r T a D2 U 3 þ a2 Ra Pr ½ð/1 1Þ þ ð/0 /1 Þ/ ½Pr ðD2 a2 Þ rh þ a2 ð/0 /1 Þ½HfRq 1 þ bðT 1 T 0 Þg r ðD2 a2 Þ r/ ¼ 0: þ Ra Pr h½P
ð50Þ
We now consider the case where both the boundaries are free. For free–free boundaries surfaces the appropriate boundary conditions are:
U 3 ¼ 0;
D2 U 3 ¼ 0;
h ¼ 0;
/ ¼ 0 at X 3 ¼ 0 and X 3 ¼ 1:
ð51Þ
For neutral stability, the real part of r has to be zero. Hence r = iw, where w is real and represents dimensionless frequency. The Galerkin-type weighted residuals method is used to obtain an approximate solution to the system of Eqs. (44), (45), (50) and the corresponding boundary conditions (51). In this method the test functions are taken as sinppX3 satisfying the boundary conditions for U3p, hp and /p. Thus U3, h and / can be assumed as:
U3 ¼
N X P¼1
Ap U 3p ; h ¼
N X P¼1
Bp hp ; / ¼
N X
C p /p ;
where Ap ; Bp
and C p are the unknown coefficients:
P¼1
Substituting above values into Eqs. (44), (45) and (50), we obtained a system of 3N linear algebraic equations in the 3N unknowns Ap, Bp, Cp, where p = 1, 2, 3, . . . , N and N is a natural number. For the existence of non trivial solution, the vanishing of the determinant of coefficients produces the eigenvalue equation for the system in terms of Rayleigh number R a, which shows that R a can be expressed in terms of the other parameters.
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4. Natural convection 4.1. Non-oscillatory convection First we consider the case of non-oscillatory instability, when w = 0. For this as a first approximation, choosing N = 1, this gives:
Ra ¼ ½2bNBT ðT 0 T 1 Þðða2 þ p2 Þ3 þ p2 T a Þ=ða2 ð2bNBT ðT 0 T 1 Þ þ /1 ð2 þ 2Rq þ bð1 þ NBT ÞðT 0 þ T 1 Þ þ NTT ð2 2Rq þ bT 0 bT 1 ÞÞ þ /0 ð2 2Rq bð1 þ NBT ÞðT 0 T 1 Þ þ NTT ð2 þ 2Rq bT 0 þ bT 1 ÞÞÞÞ:
ð52Þ
In the absence of rotation (Ta = 0), we get:
Ra ¼
2NBT ða2 þ p2 Þ3
h
i : 2ð1þRq bðT 0 T 1 ÞÞ a2 ð2 þ /0 þ /1 ÞNBT þ ð/0 /1 Þð1 NTT Þ 1 þ bðT 0 T 1 Þ
ð53Þ
This is the same expression for Rayleigh number as obtained by authors (Dhananjay, Agrawal, & Bhargava, 2011). To obtain the critical wave number, let a2 = p2x, then Eq. (52) becomes:
Ra ¼ ½2bNBT ðT 0 T 1 ÞðT a þ p4 ð1 þ xÞ3 Þ=ðð2bNBT ðT 0 T 1 Þ þ /1 ð2 þ 2Rq þ bð1 þ NBT ÞðT 0 þ T 1 Þ þ NTT ð2 2Rq þ bT 0 bT 1 ÞÞ þ /0 ð2 2Rq bð1 þ NBT ÞðT 0 T 1 Þ þ NTT ð2 þ 2Rq bT 0 þ bT 1 ÞÞÞxÞ:
ð54Þ
The critical cell size is at the onset of instability, which is obtained from the condition
dRa ¼ 0; dx x¼xc 1 ) xc ¼ þ 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3
p12 þ 2p8 T a þ 2 p20 T a þ p16 T 2a
p4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2ðp12 þ 2p8 T a þ 2 p20 T a þ p16 T 2a Þ1=3
2p4
:
ð55Þ
The corresponding critical wave number ac is obtained as:
0
B 1 p4 B ac ¼ pB þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3 þ @ 2 12 8 2 p þ 2p T a þ 2 p20 T a þ p16 T 2a
p þ 2p 12
8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3 11=2 T a þ 2 p20 T a þ p16 T 2a C C C : 4 A 2p
ð56Þ
Thus critical wave number ac is independent of the properties of nanofluid, which is dependent on the Taylor’s number. The critical Rayleigh number Rc, can be calculated from Eq. (52) with a being replaced by ac. From Eq. (56) for large Ta(Ta ? 1), critical wave number ac is obtained as:
ac ¼
1 2 p Ta 2
1=6
¼ 1:3048ðT a Þ1=6
ðT a ! 1Þ:
It is the same critical wave number for Rayleigh–Bénard instability with rotation for the regular fluid obtained by Chandrasekhar (1961) in the case of large Ta(Ta ? 1). In the absence of nanoparticles (/0 = /1 = 0), Eq. (52) gives:
Ra ¼ p4
1 Ta ð1 þ xÞ3 þ 4 ; x p
ð57Þ
this is exactly the same expression for Rayleigh number for Rayleigh–Bénard instability with rotation in the case of regular fluid as obtained by Chandrasekhar (1961). 4.2. Oscillatory convection In case of oscillatory convection (w – 0), using the one-term Galerkin approximation; the characteristic equation takes the form:
1 ½a2 Pr Ra ðJPr þ iwÞð2ð/0 /1 ÞJAC Jð2NBT þ /1 NBT /1 A þ /0 ðNBT þ AÞÞS 2iðð/0 /1 ÞC þ ð1 þ /1 ÞSÞwÞ S 2 þ 2SðJ þ iwÞðJN BT þ iwÞðP2r ðJ3 þ p2 T a Þ þ 2iJ P r w Jw2 Þ ¼ 0; where J = a2 + p2, S = b(T0 T1), A = NTT 1, C = Rq 1.
ð58Þ
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The real and imaginary part of Eq. (58) gives
J 2 P2r ð2NBT ðJ 3 þ p2 T a ÞS þ a2 Ra ðAð/0 /1 Þð2C SÞ þ ð2 þ /0 þ /1 ÞNBT SÞÞ 2ððJ 3 ðNBT þ 2ð1 þ NBT ÞPr þ P2r Þ þ P2r p2 T a ÞS þ a2 Pr Ra ðCð/0 /1 Þ 2
ð59Þ
4
þð1 þ /0 ÞSÞÞw þ 2JSw ¼ 0; 2P r ðJ 3 ðPr þ NBT ð2 þ Pr ÞÞ þ ð1 þ N BT Þp2 P r T a ÞS þ a2 Pr Ra ð2Cð/0 /1 Þ ðA þ Pr Þ þ ðAð/0 /1 Þ þ ð2 þ /0 þ /1 ÞNBT þ 2ð1 þ /1 ÞPr ÞSÞ
ð60Þ
2Jð1 þ NBT þ 2P r ÞSw2 ¼ 0: On eliminating Ra between the Eqs. (59) and (60) we have
Mk2 þ Nk þ Q ¼ 0;
where k ¼ w2
and
M ¼ JSð2Cð/0 /1 Þð1 þ A þ NBT þ P r Þ þ ð2 þ Að/0 þ /1 Þ /0 NBT 2Pr þ /1 ð2 þ NBT þ 2P r ÞÞSÞ;
ð61Þ ð62Þ
N ¼ Sð2Cð/0 /1 ÞPr ðJ 3 ðNBT ð1 þ Pr Þ þ Pr þ P 2r Þ þ p2 Pr ð1 NBT þ Pr ÞT a Þ Að/0 /1 ÞðJ3 ðNBT þ Pr þ NBT Pr P2r Þ þ p2 P2r T a Þð2C SÞ þ ðJ 3 ðð/0 /1 ÞNBT ð1 þ Pr ÞPr þ ð2 þ /0 þ /1 ÞN2BT ð1 þ Pr Þ þ 2ð1 þ /1 ÞP2r ð1 þ Pr ÞÞ þ p2 ð2 þ /0 NBT /1 ð2 þ NBT 2P r Þ 2Pr ÞP2r T a ÞSÞ;
ð63Þ
Q ¼ J 2 P2r Sð2Cð/0 /1 ÞNBT Pr ðJ 3 þ p2 T a Þ Að/0 /1 ÞðJ 3 ðN BT þ P r þ NBT Pr Þ þ p2 ðNBT ð1 þ Pr Þ þ Pr ÞT a Þð2C SÞ þ NBT ðJ 3 ðð/0 /1 ÞP r þ ð2 þ /0 þ /1 ÞNBT ð1 þ Pr ÞÞ þ p2 ðð2 þ /0 þ /1 ÞNBT ð1 þ Pr Þ þ ð/0 /1 ÞPr ÞT a ÞSÞ:
ð64Þ
2
Elimination of w , between the Eqs. (59) and (60) gives:
Pr ð8ð1 þ NBT ÞðJ 3 ð1 þ P r Þ2 þ p2 P 2r T a ÞðJ 3 ðNBT þ Pr Þ2 þ p2 P2r T a ÞS2 þ 2a2 Ra Sð2Cð/0 /1 ÞPr ðJ 3 ð1 þ Pr Þð3N2BT þ Pr ð3 þ 2Pr Þ þ NBT ð3 þ 5Pr ÞÞ þ p2 Pr ð1 þ N2BT þ Pr þ 2P 2r þ NBT ð2 þ Pr ÞÞT a Þ Að/0 /1 Þð1 þ NBT 2Pr ÞðJ 3 ð1 þ P r ÞðN BT þ Pr Þ p2 P2r T a Þð2C SÞ þ ðJ 3 ð1 þ Pr ÞðNBT þ Pr Þðð2 þ /0 þ /1 ÞN2BT þ 2ð1 þ /1 ÞPr ð3 þ 2Pr Þ þ NBT ð2 þ /0 þ /1 2/0 Pr þ 4/1 Pr ÞÞ þ p2 P2r ð/0 N2BT NBT ð2 þ /0 þ 6Pr 2/0 P r Þ 2ð1 þ Pr þ 2P 2r Þ þ /1 ð2 þ 3NBT þ N2BT þ 2Pr þ 4NBT Pr þ 4P2r ÞÞT a ÞSÞ þ a4 Pr R2a ð4C 2 ð/0 /1 Þ2 Pr ð1 þ NBT þ Pr Þ þ 2Cð/20 NBT ð1 þ NBT Þ 2/0 ðN2BT 2ð1 þ /1 ÞPr ð1 þ Pr Þ þ NBT ð1 þ 2Pr 2/1 P r ÞÞ /1 ðð2 þ /1 ÞN 2BT þ 4ð1 þ /1 ÞPr ð1 þ Pr Þ þ NBT ð2 þ /1 4Pr þ 4/1 Pr ÞÞÞS þ ðð2/0 /20 þ ð2 þ /1 Þ/1 ÞN2BT þ 4ð1 þ /1 Þ2 P r ð1 þ P r Þ þ 2ð1 þ /1 ÞNBT ð2 þ /0 þ /1 2Pr þ 2/1 Pr ÞÞS2 A2 ð/0 /1 Þ2 ð2C þ SÞ2 2Að/0 /1 Þð2C SÞfCð/0 /1 Þð1 þ NBT Þ þ ð1 þ /1 þ NBT /0 NBT ÞSgÞÞ ¼ 0:
ð65Þ
On solving (61), angular frequency w of the oscillation is given by:
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 14 N N Q k¼w ¼ 4 5: 2 M M M 2
ð66Þ
Case 1: M – 0: Since w is real for overstability, both values of k (=w2) should be positive. In order that w to be real; the sum of the roots of Eq. (61) should be positive, i.e.
N P 0: M
ð67aÞ
This is the necessary condition for occurrence of oscillatory convection. On using the Descartes’ rule of sign the necessary and sufficient conditions for occurrence of oscillatory convection are the following:
8 2 Q N > < M 4 M P 0; N M P 0; > :Q P 0; M
ð67bÞ
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1179
Case 2: M = 0 and N – 0: From Eq. (61), k ¼ QN , in order that x to be real, k(=x2) should be positive. Therefore, the necessary and sufficient conditions for occurrences of oscillatory convection is
Q P 0: N
ð67cÞ
Hence, Eq. (65) gives the oscillatory stability boundary condition under the condition (67a) and (67b) to be true. However, in the absence of nanoparticles i.e. for regular fluids (/0 = /1 = 0), Eqs. (67a) and (65) become:
Fig. 2. Effect of Taylor number Ta on Rayleigh number Ra for alumina-water-based nanofluid.
Fig. 3. Effect of Taylor number Ta on the Rayleigh number Ra for copper-water nanofluid.
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Fig. 4. Variation of critical Rayleigh number Rc with Taylor number Ta.
Table 1 Non oscillatory convection result for variation of Taylor number Ta of Al2O3 water nanofluid on critical Rayleigh number Rc. Ta
ac
Rc
Ta
ac
Rc
0 10 102 200 300 400 500 103 2 103 5 103 104
2.22144 2.270 2.594 2.832 3.011 3.155 3.278 3.710 4.221 5.011 5.698
48.8853 50.3399 61.4337 71.3756 79.9614 87.6742 94.7627 124.6175 170.9324 272.8439 399.7845
3 104 105 3 105 106 107 108 109 1010 1011 1012 1013
6.691 8.626 10.45 12.86 19.02 28.02 41.20 60.52 88.87 130.46 191.51
758.7475 1.5843 103 3.1653 103 6.8567 103 3.0832 104 1.4104 105 6.5027 105 3.0089 106 1.3946 107 6.4689 107 3.0016 108
Table 2 Non oscillatory convection result for variation of Taylor number Ta of Cu water nanofluid on critical Rayleigh number Rc. Ta
ac
Rc
Ta
ac
Rc
0 10 102 200 300 400 500 103 2 103 5 103 104
2.221 2.270 2.594 2.832 3.011 3.155 3.278 3.710 4.221 5.011 5.698
19.0325 19.5989 23.9180 27.7887 31.1314 34.1343 36.8940 48.5174 66.5492 106.2264 155.6483
3 104 105 3 105 106 107 108 109 1010 1011 1012 1013
6.691 8.626 10.45 12.86 19.02 28.02 41.20 60.52 88.87 130.46 191.51
295.4035 616.8165 1.2323 103 2.6695 103 1.2004 104 5.4912 104 2.5317 105 1.1715 106 5.4296 106 2.5185 107 1.1686 108
T 1 1 Pr ð1 þ xÞ2 P 0; ð1 þ xÞ 1 þ Pr " # 2ð1 þ Pr Þ P2r 3 R1 ¼ ð1 þ xÞ þ T 1 ; x ð1 þ Pr Þ2
ð68Þ ð69Þ
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Fig. 5. Effect of volumetric fraction on the Rayleigh number.
Fig. 6. Effect of temperature difference on the Rayleigh number.
where
x¼
a2
p2
; w1 ¼
w
p2
; R1 ¼
R
p4
; T1 ¼
Ta
p2
:
ð70Þ
Eqs. (68) and (69) are the same equations as obtained by Chandrasekhar (1961) for regular fluid. These equations must be satisfied if overstability has to occur for a wave number corresponding to x and Ta value corresponding to T1. 5. Results and discussion Here we now provide a detailed discussion of the role played by various nanofluid parameters and effect of rotation on the flow for stability analysis. According to Tzou (2008a, 2008b) for alumina-water nanofluid thermo fluid characteristic are:
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Fig. 7. Effect of Rq on the Rayleigh number.
Fig. 8. Effect of NBT on the Rayleigh number.
Table 3 Non oscillatory convection result for variation of volumetric fraction of nanoparticle on critical Rayleigh number Rc. Ta = 0
Ta = 10
Ta = 100
D/
ac
Rac
D/
ac
Rac
D/
ac
Rac
0 0.005 0.01 0.02 0.03 0.04 0.05
2.2214 2.2214 2.2214 2.2214 2.2214 2.2214 2.2214
657.5114 41.6879 21.5200 10.9374 7.3319 5.5141 4.4187
0 0.005 0.01 0.02 0.03 0.04 0.05
2.2701 2.2701 2.2701 2.2701 2.2701 2.2701 2.2701
677.0769 42.9284 22.1604 11.2628 7.5500 5.6782 4.5501
0 0.005 0.01 0.02 0.03 0.04 0.05
2.5935 2.5935 2.5935 2.5935 2.5935 2.5935 2.5935
826.2896 52.3888 27.0440 13.7449 9.2139 6.9296 5.5529
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DT ¼ 10K;
D/ ¼ /0 /1 ¼ 0:01;
/1 ¼ 0:009;
RN ¼ 30:18;
NBT ¼ 0:2;
b ¼ 6 103 ;
Rq ¼ 4
and for copper-water nanofluid thermo fluid characteristics are:
DT ¼ 10K;
D/ ¼ /0 /1 ¼ 0:01;
/1 ¼ 0:009;
RN ¼ 3:018;
NBT ¼ 2;
b ¼ 6 104 ;
Rq ¼ 9:
The above calculations are based on the 10 nm nanoparticles. Figs. 2 and 3 show the variation of Rayleigh number for alumina-water and copper-water nanofluids for different values of Taylor’s number respectively. Fig. 4 shows the variation of
Table 4 Non oscillatory convection result for variation of temperature difference of nanoparticle on critical Rayleigh number Rc. Ta = 0
Ta = 10
Ta = 100
DT
ac
Rac
DT
ac
Rac
DT
ac
Rac
20 40 60 80 100 120 140
2.2214 2.2214 2.2214 2.2214 2.2214 2.2214 2.2214
5.4960 10.9136 16.2544 21.5200 26.7120 31.8320 36.8813
20 40 60 80 100 120 140
2.2701 2.2701 2.2701 2.2701 2.2701 2.2701 2.2701
5.6596 11.2384 16.7381 22.1604 27.5059 32.7792 37.9788
20 40 60 80 100 120 140
2.5935 2.5935 2.5935 2.5935 2.5935 2.5935 2.5935
6.9068 13.7151 20.4268 27.0440 33.5688 40.0030 46.3485
Table 5 Non oscillatory convection result for variation of Rq on critical Rayleigh number Rc. Ta = 0
Ta = 10
Ta = 100
Rq
ac
Rac
Rq
ac
Rac
Rq
ac
Rac
2 3 4 5 6 7 8 9
2.2214 2.2214 2.2214 2.2214 2.2214 2.2214 2.2214 2.2214
96.7102 51.6203 35.2060 26.7120 21.5200 18.0179 15.4961 13.5935
2 3 4 5 6 7 8 9
2.2701 2.2701 2.2701 2.2701 2.2701 2.2701 2.2701 2.2701
99.5880 53.1564 36.2536 27.5069 22.1604 18.5540 15.0572 13.9980
2 3 4 5 6 7 8 9
2.5935 2.5935 2.5935 2.5935 2.5935 2.5935 2.5935 2.5935
121.5350 64.8708 44.2431 33.5688 27.0440 22.6429 19.4738 17.0829
Table 6 Non oscillatory convection result for variation of RN on critical Rayleigh number Rc. Ta = 0
Ta = 10
Ta = 100
RN
ac
Rac
RN
ac
Rac
RN
ac
Rac
10 20 30 40 50 60 70
2.2214 2.2214 2.2214 2.2214 2.2214 2.2214 2.2214
95.8859 35.3507 21.6699 15.6236 12.2153 10.0277 8.5047
10 20 30 40 50 60 70
2.2701 2.2701 2.2701 2.2701 2.2701 2.2701 2.2701
98.7392 36.4026 22.3147 16.0885 12.5787 10.3261 8.7577
10 20 30 40 50 60 70
2.5935 2.5935 2.5935 2.5935 2.5935 2.5935 2.5935
120.4991 44.4249 27.2324 19.6340 15.3508 12.6017 10.6877
Table 7 Non oscillatory convection result for NBT on critical Rayleigh number Rc. Ta = 0
Ta = 10
Ta = 100
NBT
ac
Rac
NBT
ac
Rac
NBT
ac
Rac
0.2 0.5 0.7 1.0 1.5 2 3
2.2214 2.2214 2.2214 2.2214 2.2214 2.2214 2.2214
21.5200 19.2953 18.9227 18.6526 18.4477 18.3470 18.2473
0.2 0.5 0.7 1.0 1.5 2 3
2.2701 2.2701 2.2701 2.2701 2.2701 2.2701 2.2701
22.1604 19.8695 19.4858 19.2076 18.99967 18.8929 18.7903
0.2 0.5 0.7 1 1.5 2 3
2.5935 2.5935 2.5935 2.5935 2.5935 2.5935 2.5935
27.0440 24.2483 23.7800 23.4405 23.1831 23.0565 22.9313
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critical Rayleigh number Rc versus Taylor number Ta for Alumina-water and Copper-water nanofluids. These figures indicate that the critical Rayleigh-number increases as Taylor-number increases showing that the rotation has a stabilizing effect. From the Tables 1 and 2, we obtained that in the absence of rotation (Ta = 0), the critical Rayleigh number for Cu-water nanofluid is 19.0325 and for Al2O3-water nanofluid is 48.8853.This is the same result as obtained by authors (Dhananjay et al., 2011). Also we have seen that the critical Rayleigh number for Cu-water nanofluid is lower in comparison with the critical Rayleigh number for Al2O3-water nanofluid. It can be concluded that the Al2O3-water nanofluid is more stable than Cu-water nanofluid. Threshold values for water nanofluids with metallic/metallic oxide nanoparticles of 1 100 nm are taken as follows: RN ¼ 30:18;
N BT ¼ 0:2;
b ¼ 5:30 104
1 ; K
Rq ¼ 6;
DT ¼ T 0 T 1 ¼ 80K;
T a ¼ 100;
D/ ¼ 0:01;
/1 ¼ 0:009:
We have used these values in the Figs. 5–8 and Tables 3–7. Fig. 5 represents the variation of Rayleigh number versus wave number for the different values of volumetric fraction. It is observed that the volumetric fraction has destabilizing effect. This may lead to an increase in volumetric fraction, which shows that Brownian motion of the nanoparticles will also increase, which may cause destabilizing effect. From the Table 3, we obtained that in the absence of nanoparticles (D/ = /0 /1 = 0) the critical Rayleigh numbers remain same for different values of Taylor number as obtained by Chandrasekhar (1961). We have plotted the Rayleigh number for different values of temperature difference in Fig. 6. As temperature difference increases, Buoyancy forces also increases which stabilizes the system as observed from the Fig. 6. The behavior of density ratio Rq has been shown been in Fig. 7 keeping other parameters as constant. It has destabilizing effect because the heavier nanoparticles moving through the base fluid makes more strong disturbances as compared with the lighter nanoparticles. Fig. 8 shows the variation of Rayleigh number for the different values of non-dimensional parameter NBT and concludes that it has destabilized the system. 6. Conclusion A linear analysis of Rayleigh–Bénard convection in rotating nanofluids is investigated in this work. We have used the Galerkin-type weighted residuals method for the stability analysis. The behavior of various parameters like rotation, differences of the temperature, volume concentration between the two plates, the thermal expansion of the nanofluid, density ratio of the nanoparticles to the base fluid and the Brownian to thermal diffusivity ratio on the onset of convection has been analysed analytically and numerically. Result has been depicted graphically. The result shows that for the case of non oscillatory convection rotation and difference on the temperature has a stabilizing effect. The joint behavior of Brownian motion and thermoporesis of nanoparticles creates destabilizing effect, which can reduce the critical Rayleigh number by as much as one to two orders of magnitude as compared to that of regular fluids without nanoparticles. The necessary and sufficient conditions for the existence of over stability are also derived. Acknowledgement The authors are thankful to the referee for valuable suggestions; leading to the overall improvements in the paper. References Ahmadi, G. (1976). Stability of a micro polar fluid layer heated from below. International Journal of Engineering Science, 14, 81–89. Alloui, Z., Vasseur, P., & Reggio, M. (2010). Natural convection of nanofluids in a shallow cavity heated from below. International Journal of Thermal Science, 50(3), 385–393. Buongiorno, J. (2006). Convective transport in nanofluids. ASME Journal of Heat Transfer, 128, 240–250. Chandrasekhar, S. (1961). Hydrodynamic and hydromagnetic stability. Oxford: Oxford University press. Choi, S. (1995). Enhancing thermal conductivity of fluids with nanoparticles. In Siginer, D.A., Wang, H.P., (Eds). Developments and applications of NonNewtonian flows, ASME FED 231/MD Vol. 66 (pp. 99–105). Dhananjay Agrawal, G. S., & Bhargava, R. (2011). Rayleigh–Bénard convection in nanofluid. International Journal of Applied Mathamatics and Mechanics, 7(2), 61–76. Kim, J., Kang, Y. T., & Choi, C. K. (2004). Analysis of convective instability and heat transfer characteristics of nanofluids. Physics of Fluid, 16, 2395–2401. Nield, D. A., & Kuznetsov, A. V. (2010). The onset of convection in a horizontal nanofluid layer of finite depth. European Journal of Mechanics B/Fluids, 29, 217–233. Qin, Y., & Kaloni, P. N. (1992). A thermal instability problem in a rotating micropolar fluid. International Journal of Engineering Science, 30, 1117–1126. Rajagopal, K. R., Saccomandi, G., & Vergori, L. (2009). On the Oberbeck- Boussinesq approximation in fluids with pressure-dependent viscosities. Nonlinear Analysis: Real World Applications, 10, 1139–1150. Tzou, D. Y. (2008a). Thermal instability of nanofluids in natural convection. International Journal of Heat and Mass Transfer, 51, 2967–2979. Tzou, D. Y. (2008b). Instability of nanofluids in natural convection. ASME Journal of Heat Transfer, 130, 372–401.
Contents lists available at SciVerse ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Thermal instability of rotating nanofluid layer Dhananjay Yadav ⇑, G.S. Agrawal, R. Bhargava Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India
a r t i c l e
i n f o
Article history: Received 18 November 2010 Received in revised form 2 July 2011 Accepted 13 July 2011 Available online 23 August 2011 Keywords: Nanofluid Brownian motion Thermal instability Galerkin method Critical Rayleigh number Thermophoresis
a b s t r a c t In the present paper we have considered thermal instability of rotating nanofluids heated from below. Linear stability analysis has been made to investigate analytically the effect of rotation. The more important effect of Brownian motion and thermophoresis has been included in the model of nanofluid. Galerkin method is used to obtain the analytical expression for both non-oscillatory and oscillatory cases, when boundaries surfaces are free–free. The influence of various nanofluids parameters and rotation on the onset of convection has been analysed. It has been shown that the rotation has a stabilizing effect depending upon the values of various nanofluid parameters. The critical Rayleigh number for the onset of instability is determined numerically and results are depicted graphically. The necessary and sufficient conditions for the existence of over stability are also obtained. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Nanofluid is the suspension of nanoparticles in a base fluid, which was first utilized by Choi (1995). Nanoparticles used in nanofluid preparation usually have diameters below 100 nm. Due to their small size, nanoparticles fluidize easily inside the base fluid, and as a consequence, blockage of channels and erosion in channel walls are no longer a problem. Nanoparticles materials include oxide ceramics (Al2O3, CuO), metal carbides (SiC), nitrides (AlN, SiN), metals (Al, Cu) etc. As mentioned in the literature, base fluid include water, ethylene or tri-ethylene-glycols and other coolants, oil and other lubricants, biofluids, polymer solutions, other common fluids. In order to improve the stability of nanoparticles inside the base fluid, some additives are added to the mixture in small amounts. Buongiorno (2006) studied the convective transport in nanofluids and observed that the absolute velocity of the nanoparticles can be viewed as the sum of the base fluid velocity and a relative velocity. He also discussed the effect of seven slip mechanisms: inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage, and gravity setting. He concluded that in the absence of turbulent eddies, Brownian diffusion and thermophoresis dominates the other slip mechanisms. The Rayleigh–Bénard problem for a regular fluid with rotation or without rotation was first discussed by Chandrasekhar (1961). Stability of a micro polar fluid layer heated from below has been studied by Ahmadi (1976). Qin and Kaloni (1992) have considered a thermal instability problem in rotating micro polar fluids. They obtained that for low values of Taylor number rotation has a stabilizing effect. Thermal instability problem for nanofluid without rotation was studied by Tzou (2008a, 2008b) and Dhananjay, Agarwal, and Bhargava (2011). The onset of convection in a horizontal nanofluid layer of finite depth was studied by Nield and Kuznetsov (2010). Alloui, Vasseur, and Reggio (2010) studied the natural convection of nanofluids in a shallow cavity heated from below. They observed that the presence of nanoparticles in a fluid is found to reduce the strength of flow field, this behavior being more ⇑ Corresponding author. Tel.: +91 9286685908. E-mail addresses: [email protected] (D. Yadav), [email protected] (G.S. Agrawal), [email protected] (R. Bhargava). 0020-7225/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2011.07.002
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Nomenclature a wave number c specific heat d diameter of nanoparticles D distance (m) DBT diffusion coefficient (m2/s) g gravitation (m/s2) hp = cprT enthalpy (j/kg) of nanoparticles j mass flux (kg/m2s) k thermal conductivity (w/mK) Boltzmann constant (j/K) kB p pressure (Pa) P pressure t time (s) ui velocity (m/s), i = 1, 2, 3 Ui velocity,i = 1, 2, 3 xi space (m), i = 1, 2, 3 coordinate, i = 1, 2, 3 Xi Pr Prantdl number Ra Rayleigh number Greek symbols / volume fraction of nanoparticles a thermal diffusivity (m2/s) b thermal expansion coefficients (1/K) q mass density (kg/m3) d Kronecker delta Z x3-component of vorticity. s time
pronounced at low Rayleigh number. Also, the temperatures on the solid boundaries are reduced (enhanced) by the presence of the nanoparticles when the strength of convection is high (low). For completeness a substantially different treatment of Rayleigh- Bénard problem for nanofluids was given by Kim, Kang, and Choi (2004). In this paper the quantities namely the thermal expansion coefficients, the thermal diffusivity, and kinematic viscosity that appear in the definition of Rayleigh number were modified with respect to the nanofluid. The effect of rotation on the Rayleigh–Bénard convection in nanofluids is important in certain chemical engineering and biochemical engineering. Rayleigh–Bénard convection in rotating nanofluids about a vertical axis combines the element of thermal buoyancy and rotation induced Coriolis and centrifugal forces. Due to the Coriolis force on the Rayleigh–Be‘nard convection another parameter namely Taylor number is introduced in this problem. Taylor number is a non-dimensional number which is a measure of rotation rate. It is apparent that Rayleigh–Bénard convection in rotating nanofluids will play an important role in many physical phenomenon concerning with geophysics, astrophysics and oceanography. In the present paper we have studied the thermal instability of rotating nanofluid layer with free boundaries. We have obtained the critical Rayleigh number as well as critical wave number by using Galerkin-type weighted residuals method. Stability is discussed analytically as well as numerically. It has been observed that the temperature gradient and rotation have stabilizing effect, while volumetric fraction of nanoparticles and the density ratio of nanoparticles vis-a-vis base fluid have destabilizing effect on the system. We have also discussed the case of over stability and compared our results with that of Chandrasekhar (1961). 2. Problem formulation Consider an infinite horizontal layer of incompressible nanofluid which is kept rotating about vertical axis at a constant angular velocity X = (0, 0, X) and heated from below. Let us consider the Cartesian co-ordinate system x1, x2, x3 in which x3 is taken at right angle to the boundaries. The nanofluid is confined between two parallel plates x3 = 0 and x3 = L, where temperature and volumetric fraction of nanoparticles are kept constants: T = T0 and / = /0 at x3 = 0 and T = T1, / = /1 at x3 = L (shown in the Fig. 1). We assume that the both boundaries surfaces are free. We have taken the thermo-physical properties of nanofluids (viscosity, density, thermal conductivity, and specific heat) as constants for the analytical formulation but these quantities are not constant and strongly depend on the volume fraction of nanoparticles. Under the Boussinesq approximation (Rajagopal, Saccomandi, & Vergori, 2009), the continuity and momentum equations are
D. Yadav et al. / International Journal of Engineering Science 49 (2011) 1171–1184
1173
x3 axis Ω = (0,0, Ω)
x3 = L, φ = φ1
T1
- - - - - - - - - - - - - - - - - - - - - - - - - - - --- g = (0,0, − g ) - Incompressible
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- -
rotating nanofluid - - - - - - -- - - x2 axis
-- - - - - - - - - -- - - - - - - - - - - - -
T0
O
x1 axis
x3 = 0, φ = φ0
Fig. 1. Sketch of the problem geometry and coordinates.
r:V ¼ 0;
@ q0 þ ðV:rÞ V ¼ rp þ lr2 V qgk þ 2q0 V X; @t
ð1Þ ð2Þ
where
q0 /0 ¼ /qp þ ð1 /Þqf0 ; q ¼ /qp þ ð1 /Þqf ¼ /qp þ ð1 /Þqf0 ½1 bðT T 0 Þ;
ð3Þ
q0 and qf0 being the nanofluid density and base fluid density at reference temperature T0 respectively.
Taking the density of the nanofluid as that of the base fluid, as adopted by Tzou (2008a, 2008b), the specific weight (qg) in Eq. (2) thus becomes
qg ffi ½/qp þ ð1 /Þ½q0 f1 bðT T 0 Þgg:
ð4Þ
Due to Brownian motion and thermoporesis, the mass flux (jp) of the nanoparticles in base fluid is given by
jp ¼ qp DB r/ qp
DT rT; T
ð5Þ
where DB is the Brownian diffusion coefficient, given by Einstein-Stokes equation, and DT is the thermoporetic diffusion coefficient of the nanoparticles given as:
DB ¼
kB T ; 3pldp
DT ¼
l q
0:26k /: 2k þ kp
ð6Þ
In the linear theory, temperature change in the nano fluid is less compared to the reference temperature (T0) (Tzou, 2008a, 2008b). Therefore, nanofluid temperature T in the Eq. (5) has been replaced by T0. The continuity equation for the nanoparticles is given by:
@ 1 DT þ ðV:rÞ / ¼ r:jp ¼ DB rðr/Þ þ rðrTÞ: @t qp T0
ð7Þ
The energy equation is:
q0 c
@ þ ðV:rÞ T ¼ r:q þ hp r:jp ; @t
ð8Þ
where hpr.jp is the addition flow work in energy equation due to Brownian motion and thermophoresis of nanoparticles relative to the flow velocities (Buongiorno, 2006) and
q ¼ krT þ hp jp ) r:q ¼ kr2 T þ r:ðhp jp Þ ¼ kr2 T þ hp r:jp þ jp :rhp ¼ kr2 T þ hp r:jp þ jp :ðcp rTÞ; where rhp is equal to cprT. Substituting the expression for r.q and jp, the energy Eq. (8) becomes as:
q0 c
@ DT ðrTÞ2 ; þ ðV:rÞ T ¼ krðrTÞ þ qp cp DB ðr/ÞðrTÞ þ @t T0
and vorticity is given by:
ð9Þ
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D. Yadav et al. / International Journal of Engineering Science 49 (2011) 1171–1184
n ¼ r V:
ð10Þ
Eqs. (1), (2), (7), (9) and (10) gives seven equations for seven unknown: three velocity components (u1, u2, u3), pressure (p), volumetric fraction of nanoparticles (/), temperature (T) and x3- component of vorticity. Introducing the following non dimensional parameters
Xi ¼
xi ; L
s¼
t L2
a; P ¼
p L2 ; ðq0 a2 Þ
/¼
/ /1 ; /0 /1
h¼
T T1 ; T0 T1
Ui ¼
ui L; ðaÞ
Z¼
n L2
a;
ð11Þ
the non dimensional form of Eqs. (1), (2), (7), (9) and (10) become as:
@U 1 @U 2 @U 3 þ þ ¼ 0; @X 1 @X 2 @X 3 pffiffiffiffiffi @U 1 @U 1 @U 1 @U 1 @P þ U1 þ U2 þ U3 ¼ þ P r r2 U 1 þ P r T a U 2 ; @X 1 @s @X 1 @X 2 @X 3 pffiffiffiffiffi @U 2 @U 2 @U 2 @U 2 @P þ U1 þ U2 þ U3 ¼ þ P r r2 U 2 P r T a U 1 ; @X 2 @s @X 1 @X 2 @X 3
ð12Þ ð13Þ ð14Þ
@U 3 @U 3 @U 3 @U 3 @P þ U1 þ U2 þ U3 ¼ þ Pr r2 U 3 H½f1 þ bðT 1 T 0 Þgð/1 1Þ þ Rq /1 Ra Pr ð/1 1Þh @X 3 @s @X 1 @X 2 @X 3 H½Rq 1 þ bðT 1 T 0 Þð/0 /1 Þ/ Ra Pr ð/0 /1 Þ/h; with H ¼
ð15Þ
Ra P r bðT 0 T 1 Þ
@/ @/ @/ @/ þ U1 þ U2 þ U3 ¼ N BT r2 / þ NTT r2 h; @s @X 1 @X 2 @X 3
ð16Þ
@h @h @h @h 1 @h @/ @h @/ @h @/ þ þ þ U1 þ U2 þ U3 ¼ r2 h þ @s @X 1 @X 2 @X 3 Le @X 1 @X 1 @X 2 @X 2 @X 3 @X 3 " 2 2 2 # RN @h @h @h ; þ þ þ @X 1 @X 2 @X 3 Le
ð17Þ
and
Z¼
@U 2 @U 1 : @X 1 @X 2
ð18Þ
In the terms of the non-dimensional form, the boundary condition becomes as:
h ¼ 1 and / ¼ 1 at X 3 ¼ 0
ð19Þ
h ¼ 0 and / ¼ 0 at X 3 ¼ 1: The nanofluid is thus characterized by eight parameters:
Ra ¼ gL3 b Ta ¼ RN ¼
4X2
m2
ðT 0 T 1 Þ
am
ðRayleigh numberÞ;
L4 ðTaylor numberÞ;
NTT ; NBT
NTT ¼
Pr ¼
Le ¼
k
qp cp DB ð/0 /1 Þ
ðLewis numberÞ;
t DB ðPrantdl numberÞ; NBT ¼ ; a a
Rq ¼
qp ; q0 ð20Þ
DT ðT 0 T 1 Þ : T 0 a ð/0 /1 Þ
2.1. Primary flow Assuming a time-independent solution of Eqs. (12)–(18), the primary flow is supposed to be stationary i.e. U i ¼ 0 for are varying in X3 direction only. From the Eqs. (15)–(17), i = 1,2,3, whereas temperature ð hÞ and nanoparticles fraction ð/Þ the equations governing the primary flow are: 2 d /
þ RN 2
2 d h
¼ 0; 2 dX 3 2 1 d/ dh RN dh þ þ ¼ 0; 2 dX 3 Le dX 3 dX 3 Le dX 3 dX 3 2 d h
ð21Þ ð22Þ
D. Yadav et al. / International Journal of Engineering Science 49 (2011) 1171–1184
1175
@P þ Ra Pr ð/ / Þh/: ¼ H½f1 þ bðT 1 T 0 Þgð/1 1Þ þ Rq /1 þ Ra P r ð/1 1Þh þ H½Rq 1 þ bðT 1 T 0 Þð/0 /1 Þ/ 0 1 @X 3 ð23Þ
3 Þ are The boundary condition for hðX 3 Þ and /ðX
hð0Þ ¼ 1; /ð0Þ ¼ 1 and hð1Þ ¼ 0; /ð1Þ ¼ 0:
ð24Þ
Integrating Eq. (21) with respect to X3 we have
¼ RN h þ c1 X 3 þ c2 ; /
ð25Þ
where c1 and c2 are integration constants. Substituting Eq. (25) into the Eq. (22), we obtain 2 d h 2 dX 3
2
þ
c1 d h ¼ 0: Le dX 23
ð26Þ
From Eqs. (24)–(26) we have,
" hðX 3 Þ ¼
RN exp
n
ð1þRN ÞX 3 Le
o
þ exp
# 1þRN f1 ð1 þ RN ÞX 3 g Le
exp
þð1 þ RN ÞðX 3 1Þ
1 1 þ RN 1 ; Le
1 1 þ RN 3 Þ ¼ 1 exp ð1 þ RN Þð1 X 3 Þ 1 exp : /ðX Le Le
ð27Þ
ð28Þ
According to Buongiorno (2006), for most of the nanofluids RN 1 10,Le 104 105 and e = (1 + RN)/Le 105 104. in power series of e, we have, Expanding h and /
hðX 3 Þ ¼ 1 X 3 þ RN X 3 ðX 3 1Þ e þ ; 2 X ð1 X3Þ 3 3Þ ¼ 1 X3 þ /ðX e þ ; 2
ð29Þ ð30Þ
Hence, we have with e 104, as compared to (1 X3) 100, the zeroth order terms are dominant in both h and /. hðX 3 Þ ¼ /ðX 3 Þ 1 X 3 , which display linear distributions in X3. 2.2. Equations after perturbation For small disturbances onto the primary flow, we assume that
U i ¼ U i þ U 0i ;
q ¼ q þ q0 ; P ¼ P þ P 0 ; h ¼ h þ h0 ; Z ¼ Z þ Z0 and / ¼ / þ /0 ;
ð31Þ
3Þ 1 X3. with hðX 3 Þ ¼ /ðX Substituting Eq. (31) into Eqs. (12)–(18) and neglecting the product of prime quantities, we have
@U 01 @U 02 @U 03 þ þ ¼ 0; @X 1 @X 2 @X 3 pffiffiffiffiffi @U 01 @P0 ¼ þ Pr r2 U 01 þ Pr T a U 02 ; @s @X 1 pffiffiffiffiffi @U 02 @P0 ¼ þ P r r2 U 02 Pr T a U 01 ; @s @X 21 @U 03 @P0 0 Þ; ¼ þ Pr r2 U 03 Ra Pr ð/1 1Þh0 ð/0 /1 Þ½HfRq 1 þ bðT 1 T 0 Þg/0 þ Ra P r ðh/0 þ /h @s @X 3 @/0 U 03 ¼ N BT r2 /0 þ NTT r2 h0 ; @s 0 0 @h0 1 @/ @h 2RN @h0 þ ; U 03 ¼ r2 h0 Le @X 3 @s @X 3 Le @X 3 0 0 @U 2 @U 1 : Z0 ¼ @X 1 @X 2 The above system will be used in further analysis.
ð32Þ ð33Þ ð34Þ ð35Þ ð36Þ ð37Þ ð38Þ
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3. Normal mode analysis Assuming the perturbation quantities of the form as:
0 0 0 0 0 0 0 U 1 ; U 2 ; U 3 ; P ; h ; / ; Z ¼ ½U 1 ; U 2 ; U 3 ; P; h; /; ZðX 3 Þ exp½ikX 1 X 1 þ ikX2 X 2 þ rs;
ð39Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 are the wave number along the X1 and X2 directions respectively, and a ¼ kX 1 þ kX 2 is the resultant wave
where kX 1 ; kX 2 number. Following the normal mode analysis, the linearized equations in dimensionless form are as follows:
U 1 ikX 1 þ U 2 ikX 2 þ DU 3 ¼ 0;
ð40Þ
pffiffiffiffiffi þ P r ðD a ÞU 1 þ Pr T a U 2 ; pffiffiffiffiffi þ P r ðD2 a2 ÞU 2 Pr T a U 1 ; 2
2
rU 1 ¼ PikX1 rU 2 ¼ PikX2 rU 3 ¼ DP þ Pr ðD2 a2 ÞU 3 Ra Pr ð/1 1Þh ð/0 /1 Þ½HfRq 1 þ bðT 1 T 0 Þg/ þ Ra Pr ðh/ þ /hÞ;
ð42Þ ð43Þ
r/ U 3 ¼ NBT ðD2 a2 Þ/ þ NTT ðD2 a2 Þh; rh U 3 ¼ ðD2 a2 Þh
ð41Þ
ð44Þ
1 2RN ðDh þ D/Þ Dh; Le Le
ð45Þ
Z ¼ ikX 1 U 2 ikX 2 U 1 ;
ð46Þ
where D dXd 3 . From Eqs. (40)–(42) and (46) the expression for P can be obtained as:
pffiffiffiffiffi Pr ðD2 a2 ÞDU 3 rDU 3 Pr T a Z P¼ : a2
ð47Þ
Substituting Eq. (47) in Eq. (43), we have:
pffiffiffiffiffi T a DZ þ a2 Ra Pr ð/1 1Þh þ a2 ð/0 /1 Þ þ /hÞ ¼ 0: ½HfRq 1 þ bðT 1 T 0 Þg/ þ Ra Pr ðh/
ðD2 a2 Þ½Pr ðD2 a2 Þ rU 3 P r
ð48Þ
On elimination of P from Eqs. (40)–(42) and (46) we get:
½P r ðD2 a2 Þ rZ þ Pr
pffiffiffiffiffi T a DU 3 ¼ 0:
ð49Þ
From Eqs. (48) and (49), we have:
ðD2 a2 Þ½Pr ðD2 a2 Þ r2 U 3 þ P2r T a D2 U 3 þ a2 Ra Pr ½ð/1 1Þ þ ð/0 /1 Þ/ ½Pr ðD2 a2 Þ rh þ a2 ð/0 /1 Þ½HfRq 1 þ bðT 1 T 0 Þg r ðD2 a2 Þ r/ ¼ 0: þ Ra Pr h½P
ð50Þ
We now consider the case where both the boundaries are free. For free–free boundaries surfaces the appropriate boundary conditions are:
U 3 ¼ 0;
D2 U 3 ¼ 0;
h ¼ 0;
/ ¼ 0 at X 3 ¼ 0 and X 3 ¼ 1:
ð51Þ
For neutral stability, the real part of r has to be zero. Hence r = iw, where w is real and represents dimensionless frequency. The Galerkin-type weighted residuals method is used to obtain an approximate solution to the system of Eqs. (44), (45), (50) and the corresponding boundary conditions (51). In this method the test functions are taken as sinppX3 satisfying the boundary conditions for U3p, hp and /p. Thus U3, h and / can be assumed as:
U3 ¼
N X P¼1
Ap U 3p ; h ¼
N X P¼1
Bp hp ; / ¼
N X
C p /p ;
where Ap ; Bp
and C p are the unknown coefficients:
P¼1
Substituting above values into Eqs. (44), (45) and (50), we obtained a system of 3N linear algebraic equations in the 3N unknowns Ap, Bp, Cp, where p = 1, 2, 3, . . . , N and N is a natural number. For the existence of non trivial solution, the vanishing of the determinant of coefficients produces the eigenvalue equation for the system in terms of Rayleigh number R a, which shows that R a can be expressed in terms of the other parameters.
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4. Natural convection 4.1. Non-oscillatory convection First we consider the case of non-oscillatory instability, when w = 0. For this as a first approximation, choosing N = 1, this gives:
Ra ¼ ½2bNBT ðT 0 T 1 Þðða2 þ p2 Þ3 þ p2 T a Þ=ða2 ð2bNBT ðT 0 T 1 Þ þ /1 ð2 þ 2Rq þ bð1 þ NBT ÞðT 0 þ T 1 Þ þ NTT ð2 2Rq þ bT 0 bT 1 ÞÞ þ /0 ð2 2Rq bð1 þ NBT ÞðT 0 T 1 Þ þ NTT ð2 þ 2Rq bT 0 þ bT 1 ÞÞÞÞ:
ð52Þ
In the absence of rotation (Ta = 0), we get:
Ra ¼
2NBT ða2 þ p2 Þ3
h
i : 2ð1þRq bðT 0 T 1 ÞÞ a2 ð2 þ /0 þ /1 ÞNBT þ ð/0 /1 Þð1 NTT Þ 1 þ bðT 0 T 1 Þ
ð53Þ
This is the same expression for Rayleigh number as obtained by authors (Dhananjay, Agrawal, & Bhargava, 2011). To obtain the critical wave number, let a2 = p2x, then Eq. (52) becomes:
Ra ¼ ½2bNBT ðT 0 T 1 ÞðT a þ p4 ð1 þ xÞ3 Þ=ðð2bNBT ðT 0 T 1 Þ þ /1 ð2 þ 2Rq þ bð1 þ NBT ÞðT 0 þ T 1 Þ þ NTT ð2 2Rq þ bT 0 bT 1 ÞÞ þ /0 ð2 2Rq bð1 þ NBT ÞðT 0 T 1 Þ þ NTT ð2 þ 2Rq bT 0 þ bT 1 ÞÞÞxÞ:
ð54Þ
The critical cell size is at the onset of instability, which is obtained from the condition
dRa ¼ 0; dx x¼xc 1 ) xc ¼ þ 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3
p12 þ 2p8 T a þ 2 p20 T a þ p16 T 2a
p4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2ðp12 þ 2p8 T a þ 2 p20 T a þ p16 T 2a Þ1=3
2p4
:
ð55Þ
The corresponding critical wave number ac is obtained as:
0
B 1 p4 B ac ¼ pB þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3 þ @ 2 12 8 2 p þ 2p T a þ 2 p20 T a þ p16 T 2a
p þ 2p 12
8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3 11=2 T a þ 2 p20 T a þ p16 T 2a C C C : 4 A 2p
ð56Þ
Thus critical wave number ac is independent of the properties of nanofluid, which is dependent on the Taylor’s number. The critical Rayleigh number Rc, can be calculated from Eq. (52) with a being replaced by ac. From Eq. (56) for large Ta(Ta ? 1), critical wave number ac is obtained as:
ac ¼
1 2 p Ta 2
1=6
¼ 1:3048ðT a Þ1=6
ðT a ! 1Þ:
It is the same critical wave number for Rayleigh–Bénard instability with rotation for the regular fluid obtained by Chandrasekhar (1961) in the case of large Ta(Ta ? 1). In the absence of nanoparticles (/0 = /1 = 0), Eq. (52) gives:
Ra ¼ p4
1 Ta ð1 þ xÞ3 þ 4 ; x p
ð57Þ
this is exactly the same expression for Rayleigh number for Rayleigh–Bénard instability with rotation in the case of regular fluid as obtained by Chandrasekhar (1961). 4.2. Oscillatory convection In case of oscillatory convection (w – 0), using the one-term Galerkin approximation; the characteristic equation takes the form:
1 ½a2 Pr Ra ðJPr þ iwÞð2ð/0 /1 ÞJAC Jð2NBT þ /1 NBT /1 A þ /0 ðNBT þ AÞÞS 2iðð/0 /1 ÞC þ ð1 þ /1 ÞSÞwÞ S 2 þ 2SðJ þ iwÞðJN BT þ iwÞðP2r ðJ3 þ p2 T a Þ þ 2iJ P r w Jw2 Þ ¼ 0; where J = a2 + p2, S = b(T0 T1), A = NTT 1, C = Rq 1.
ð58Þ
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D. Yadav et al. / International Journal of Engineering Science 49 (2011) 1171–1184
The real and imaginary part of Eq. (58) gives
J 2 P2r ð2NBT ðJ 3 þ p2 T a ÞS þ a2 Ra ðAð/0 /1 Þð2C SÞ þ ð2 þ /0 þ /1 ÞNBT SÞÞ 2ððJ 3 ðNBT þ 2ð1 þ NBT ÞPr þ P2r Þ þ P2r p2 T a ÞS þ a2 Pr Ra ðCð/0 /1 Þ 2
ð59Þ
4
þð1 þ /0 ÞSÞÞw þ 2JSw ¼ 0; 2P r ðJ 3 ðPr þ NBT ð2 þ Pr ÞÞ þ ð1 þ N BT Þp2 P r T a ÞS þ a2 Pr Ra ð2Cð/0 /1 Þ ðA þ Pr Þ þ ðAð/0 /1 Þ þ ð2 þ /0 þ /1 ÞNBT þ 2ð1 þ /1 ÞPr ÞSÞ
ð60Þ
2Jð1 þ NBT þ 2P r ÞSw2 ¼ 0: On eliminating Ra between the Eqs. (59) and (60) we have
Mk2 þ Nk þ Q ¼ 0;
where k ¼ w2
and
M ¼ JSð2Cð/0 /1 Þð1 þ A þ NBT þ P r Þ þ ð2 þ Að/0 þ /1 Þ /0 NBT 2Pr þ /1 ð2 þ NBT þ 2P r ÞÞSÞ;
ð61Þ ð62Þ
N ¼ Sð2Cð/0 /1 ÞPr ðJ 3 ðNBT ð1 þ Pr Þ þ Pr þ P 2r Þ þ p2 Pr ð1 NBT þ Pr ÞT a Þ Að/0 /1 ÞðJ3 ðNBT þ Pr þ NBT Pr P2r Þ þ p2 P2r T a Þð2C SÞ þ ðJ 3 ðð/0 /1 ÞNBT ð1 þ Pr ÞPr þ ð2 þ /0 þ /1 ÞN2BT ð1 þ Pr Þ þ 2ð1 þ /1 ÞP2r ð1 þ Pr ÞÞ þ p2 ð2 þ /0 NBT /1 ð2 þ NBT 2P r Þ 2Pr ÞP2r T a ÞSÞ;
ð63Þ
Q ¼ J 2 P2r Sð2Cð/0 /1 ÞNBT Pr ðJ 3 þ p2 T a Þ Að/0 /1 ÞðJ 3 ðN BT þ P r þ NBT Pr Þ þ p2 ðNBT ð1 þ Pr Þ þ Pr ÞT a Þð2C SÞ þ NBT ðJ 3 ðð/0 /1 ÞP r þ ð2 þ /0 þ /1 ÞNBT ð1 þ Pr ÞÞ þ p2 ðð2 þ /0 þ /1 ÞNBT ð1 þ Pr Þ þ ð/0 /1 ÞPr ÞT a ÞSÞ:
ð64Þ
2
Elimination of w , between the Eqs. (59) and (60) gives:
Pr ð8ð1 þ NBT ÞðJ 3 ð1 þ P r Þ2 þ p2 P 2r T a ÞðJ 3 ðNBT þ Pr Þ2 þ p2 P2r T a ÞS2 þ 2a2 Ra Sð2Cð/0 /1 ÞPr ðJ 3 ð1 þ Pr Þð3N2BT þ Pr ð3 þ 2Pr Þ þ NBT ð3 þ 5Pr ÞÞ þ p2 Pr ð1 þ N2BT þ Pr þ 2P 2r þ NBT ð2 þ Pr ÞÞT a Þ Að/0 /1 Þð1 þ NBT 2Pr ÞðJ 3 ð1 þ P r ÞðN BT þ Pr Þ p2 P2r T a Þð2C SÞ þ ðJ 3 ð1 þ Pr ÞðNBT þ Pr Þðð2 þ /0 þ /1 ÞN2BT þ 2ð1 þ /1 ÞPr ð3 þ 2Pr Þ þ NBT ð2 þ /0 þ /1 2/0 Pr þ 4/1 Pr ÞÞ þ p2 P2r ð/0 N2BT NBT ð2 þ /0 þ 6Pr 2/0 P r Þ 2ð1 þ Pr þ 2P 2r Þ þ /1 ð2 þ 3NBT þ N2BT þ 2Pr þ 4NBT Pr þ 4P2r ÞÞT a ÞSÞ þ a4 Pr R2a ð4C 2 ð/0 /1 Þ2 Pr ð1 þ NBT þ Pr Þ þ 2Cð/20 NBT ð1 þ NBT Þ 2/0 ðN2BT 2ð1 þ /1 ÞPr ð1 þ Pr Þ þ NBT ð1 þ 2Pr 2/1 P r ÞÞ /1 ðð2 þ /1 ÞN 2BT þ 4ð1 þ /1 ÞPr ð1 þ Pr Þ þ NBT ð2 þ /1 4Pr þ 4/1 Pr ÞÞÞS þ ðð2/0 /20 þ ð2 þ /1 Þ/1 ÞN2BT þ 4ð1 þ /1 Þ2 P r ð1 þ P r Þ þ 2ð1 þ /1 ÞNBT ð2 þ /0 þ /1 2Pr þ 2/1 Pr ÞÞS2 A2 ð/0 /1 Þ2 ð2C þ SÞ2 2Að/0 /1 Þð2C SÞfCð/0 /1 Þð1 þ NBT Þ þ ð1 þ /1 þ NBT /0 NBT ÞSgÞÞ ¼ 0:
ð65Þ
On solving (61), angular frequency w of the oscillation is given by:
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 14 N N Q k¼w ¼ 4 5: 2 M M M 2
ð66Þ
Case 1: M – 0: Since w is real for overstability, both values of k (=w2) should be positive. In order that w to be real; the sum of the roots of Eq. (61) should be positive, i.e.
N P 0: M
ð67aÞ
This is the necessary condition for occurrence of oscillatory convection. On using the Descartes’ rule of sign the necessary and sufficient conditions for occurrence of oscillatory convection are the following:
8 2 Q N > < M 4 M P 0; N M P 0; > :Q P 0; M
ð67bÞ
D. Yadav et al. / International Journal of Engineering Science 49 (2011) 1171–1184
1179
Case 2: M = 0 and N – 0: From Eq. (61), k ¼ QN , in order that x to be real, k(=x2) should be positive. Therefore, the necessary and sufficient conditions for occurrences of oscillatory convection is
Q P 0: N
ð67cÞ
Hence, Eq. (65) gives the oscillatory stability boundary condition under the condition (67a) and (67b) to be true. However, in the absence of nanoparticles i.e. for regular fluids (/0 = /1 = 0), Eqs. (67a) and (65) become:
Fig. 2. Effect of Taylor number Ta on Rayleigh number Ra for alumina-water-based nanofluid.
Fig. 3. Effect of Taylor number Ta on the Rayleigh number Ra for copper-water nanofluid.
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D. Yadav et al. / International Journal of Engineering Science 49 (2011) 1171–1184
Fig. 4. Variation of critical Rayleigh number Rc with Taylor number Ta.
Table 1 Non oscillatory convection result for variation of Taylor number Ta of Al2O3 water nanofluid on critical Rayleigh number Rc. Ta
ac
Rc
Ta
ac
Rc
0 10 102 200 300 400 500 103 2 103 5 103 104
2.22144 2.270 2.594 2.832 3.011 3.155 3.278 3.710 4.221 5.011 5.698
48.8853 50.3399 61.4337 71.3756 79.9614 87.6742 94.7627 124.6175 170.9324 272.8439 399.7845
3 104 105 3 105 106 107 108 109 1010 1011 1012 1013
6.691 8.626 10.45 12.86 19.02 28.02 41.20 60.52 88.87 130.46 191.51
758.7475 1.5843 103 3.1653 103 6.8567 103 3.0832 104 1.4104 105 6.5027 105 3.0089 106 1.3946 107 6.4689 107 3.0016 108
Table 2 Non oscillatory convection result for variation of Taylor number Ta of Cu water nanofluid on critical Rayleigh number Rc. Ta
ac
Rc
Ta
ac
Rc
0 10 102 200 300 400 500 103 2 103 5 103 104
2.221 2.270 2.594 2.832 3.011 3.155 3.278 3.710 4.221 5.011 5.698
19.0325 19.5989 23.9180 27.7887 31.1314 34.1343 36.8940 48.5174 66.5492 106.2264 155.6483
3 104 105 3 105 106 107 108 109 1010 1011 1012 1013
6.691 8.626 10.45 12.86 19.02 28.02 41.20 60.52 88.87 130.46 191.51
295.4035 616.8165 1.2323 103 2.6695 103 1.2004 104 5.4912 104 2.5317 105 1.1715 106 5.4296 106 2.5185 107 1.1686 108
T 1 1 Pr ð1 þ xÞ2 P 0; ð1 þ xÞ 1 þ Pr " # 2ð1 þ Pr Þ P2r 3 R1 ¼ ð1 þ xÞ þ T 1 ; x ð1 þ Pr Þ2
ð68Þ ð69Þ
D. Yadav et al. / International Journal of Engineering Science 49 (2011) 1171–1184
1181
Fig. 5. Effect of volumetric fraction on the Rayleigh number.
Fig. 6. Effect of temperature difference on the Rayleigh number.
where
x¼
a2
p2
; w1 ¼
w
p2
; R1 ¼
R
p4
; T1 ¼
Ta
p2
:
ð70Þ
Eqs. (68) and (69) are the same equations as obtained by Chandrasekhar (1961) for regular fluid. These equations must be satisfied if overstability has to occur for a wave number corresponding to x and Ta value corresponding to T1. 5. Results and discussion Here we now provide a detailed discussion of the role played by various nanofluid parameters and effect of rotation on the flow for stability analysis. According to Tzou (2008a, 2008b) for alumina-water nanofluid thermo fluid characteristic are:
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Fig. 7. Effect of Rq on the Rayleigh number.
Fig. 8. Effect of NBT on the Rayleigh number.
Table 3 Non oscillatory convection result for variation of volumetric fraction of nanoparticle on critical Rayleigh number Rc. Ta = 0
Ta = 10
Ta = 100
D/
ac
Rac
D/
ac
Rac
D/
ac
Rac
0 0.005 0.01 0.02 0.03 0.04 0.05
2.2214 2.2214 2.2214 2.2214 2.2214 2.2214 2.2214
657.5114 41.6879 21.5200 10.9374 7.3319 5.5141 4.4187
0 0.005 0.01 0.02 0.03 0.04 0.05
2.2701 2.2701 2.2701 2.2701 2.2701 2.2701 2.2701
677.0769 42.9284 22.1604 11.2628 7.5500 5.6782 4.5501
0 0.005 0.01 0.02 0.03 0.04 0.05
2.5935 2.5935 2.5935 2.5935 2.5935 2.5935 2.5935
826.2896 52.3888 27.0440 13.7449 9.2139 6.9296 5.5529
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DT ¼ 10K;
D/ ¼ /0 /1 ¼ 0:01;
/1 ¼ 0:009;
RN ¼ 30:18;
NBT ¼ 0:2;
b ¼ 6 103 ;
Rq ¼ 4
and for copper-water nanofluid thermo fluid characteristics are:
DT ¼ 10K;
D/ ¼ /0 /1 ¼ 0:01;
/1 ¼ 0:009;
RN ¼ 3:018;
NBT ¼ 2;
b ¼ 6 104 ;
Rq ¼ 9:
The above calculations are based on the 10 nm nanoparticles. Figs. 2 and 3 show the variation of Rayleigh number for alumina-water and copper-water nanofluids for different values of Taylor’s number respectively. Fig. 4 shows the variation of
Table 4 Non oscillatory convection result for variation of temperature difference of nanoparticle on critical Rayleigh number Rc. Ta = 0
Ta = 10
Ta = 100
DT
ac
Rac
DT
ac
Rac
DT
ac
Rac
20 40 60 80 100 120 140
2.2214 2.2214 2.2214 2.2214 2.2214 2.2214 2.2214
5.4960 10.9136 16.2544 21.5200 26.7120 31.8320 36.8813
20 40 60 80 100 120 140
2.2701 2.2701 2.2701 2.2701 2.2701 2.2701 2.2701
5.6596 11.2384 16.7381 22.1604 27.5059 32.7792 37.9788
20 40 60 80 100 120 140
2.5935 2.5935 2.5935 2.5935 2.5935 2.5935 2.5935
6.9068 13.7151 20.4268 27.0440 33.5688 40.0030 46.3485
Table 5 Non oscillatory convection result for variation of Rq on critical Rayleigh number Rc. Ta = 0
Ta = 10
Ta = 100
Rq
ac
Rac
Rq
ac
Rac
Rq
ac
Rac
2 3 4 5 6 7 8 9
2.2214 2.2214 2.2214 2.2214 2.2214 2.2214 2.2214 2.2214
96.7102 51.6203 35.2060 26.7120 21.5200 18.0179 15.4961 13.5935
2 3 4 5 6 7 8 9
2.2701 2.2701 2.2701 2.2701 2.2701 2.2701 2.2701 2.2701
99.5880 53.1564 36.2536 27.5069 22.1604 18.5540 15.0572 13.9980
2 3 4 5 6 7 8 9
2.5935 2.5935 2.5935 2.5935 2.5935 2.5935 2.5935 2.5935
121.5350 64.8708 44.2431 33.5688 27.0440 22.6429 19.4738 17.0829
Table 6 Non oscillatory convection result for variation of RN on critical Rayleigh number Rc. Ta = 0
Ta = 10
Ta = 100
RN
ac
Rac
RN
ac
Rac
RN
ac
Rac
10 20 30 40 50 60 70
2.2214 2.2214 2.2214 2.2214 2.2214 2.2214 2.2214
95.8859 35.3507 21.6699 15.6236 12.2153 10.0277 8.5047
10 20 30 40 50 60 70
2.2701 2.2701 2.2701 2.2701 2.2701 2.2701 2.2701
98.7392 36.4026 22.3147 16.0885 12.5787 10.3261 8.7577
10 20 30 40 50 60 70
2.5935 2.5935 2.5935 2.5935 2.5935 2.5935 2.5935
120.4991 44.4249 27.2324 19.6340 15.3508 12.6017 10.6877
Table 7 Non oscillatory convection result for NBT on critical Rayleigh number Rc. Ta = 0
Ta = 10
Ta = 100
NBT
ac
Rac
NBT
ac
Rac
NBT
ac
Rac
0.2 0.5 0.7 1.0 1.5 2 3
2.2214 2.2214 2.2214 2.2214 2.2214 2.2214 2.2214
21.5200 19.2953 18.9227 18.6526 18.4477 18.3470 18.2473
0.2 0.5 0.7 1.0 1.5 2 3
2.2701 2.2701 2.2701 2.2701 2.2701 2.2701 2.2701
22.1604 19.8695 19.4858 19.2076 18.99967 18.8929 18.7903
0.2 0.5 0.7 1 1.5 2 3
2.5935 2.5935 2.5935 2.5935 2.5935 2.5935 2.5935
27.0440 24.2483 23.7800 23.4405 23.1831 23.0565 22.9313
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critical Rayleigh number Rc versus Taylor number Ta for Alumina-water and Copper-water nanofluids. These figures indicate that the critical Rayleigh-number increases as Taylor-number increases showing that the rotation has a stabilizing effect. From the Tables 1 and 2, we obtained that in the absence of rotation (Ta = 0), the critical Rayleigh number for Cu-water nanofluid is 19.0325 and for Al2O3-water nanofluid is 48.8853.This is the same result as obtained by authors (Dhananjay et al., 2011). Also we have seen that the critical Rayleigh number for Cu-water nanofluid is lower in comparison with the critical Rayleigh number for Al2O3-water nanofluid. It can be concluded that the Al2O3-water nanofluid is more stable than Cu-water nanofluid. Threshold values for water nanofluids with metallic/metallic oxide nanoparticles of 1 100 nm are taken as follows: RN ¼ 30:18;
N BT ¼ 0:2;
b ¼ 5:30 104
1 ; K
Rq ¼ 6;
DT ¼ T 0 T 1 ¼ 80K;
T a ¼ 100;
D/ ¼ 0:01;
/1 ¼ 0:009:
We have used these values in the Figs. 5–8 and Tables 3–7. Fig. 5 represents the variation of Rayleigh number versus wave number for the different values of volumetric fraction. It is observed that the volumetric fraction has destabilizing effect. This may lead to an increase in volumetric fraction, which shows that Brownian motion of the nanoparticles will also increase, which may cause destabilizing effect. From the Table 3, we obtained that in the absence of nanoparticles (D/ = /0 /1 = 0) the critical Rayleigh numbers remain same for different values of Taylor number as obtained by Chandrasekhar (1961). We have plotted the Rayleigh number for different values of temperature difference in Fig. 6. As temperature difference increases, Buoyancy forces also increases which stabilizes the system as observed from the Fig. 6. The behavior of density ratio Rq has been shown been in Fig. 7 keeping other parameters as constant. It has destabilizing effect because the heavier nanoparticles moving through the base fluid makes more strong disturbances as compared with the lighter nanoparticles. Fig. 8 shows the variation of Rayleigh number for the different values of non-dimensional parameter NBT and concludes that it has destabilized the system. 6. Conclusion A linear analysis of Rayleigh–Bénard convection in rotating nanofluids is investigated in this work. We have used the Galerkin-type weighted residuals method for the stability analysis. The behavior of various parameters like rotation, differences of the temperature, volume concentration between the two plates, the thermal expansion of the nanofluid, density ratio of the nanoparticles to the base fluid and the Brownian to thermal diffusivity ratio on the onset of convection has been analysed analytically and numerically. Result has been depicted graphically. The result shows that for the case of non oscillatory convection rotation and difference on the temperature has a stabilizing effect. The joint behavior of Brownian motion and thermoporesis of nanoparticles creates destabilizing effect, which can reduce the critical Rayleigh number by as much as one to two orders of magnitude as compared to that of regular fluids without nanoparticles. The necessary and sufficient conditions for the existence of over stability are also derived. Acknowledgement The authors are thankful to the referee for valuable suggestions; leading to the overall improvements in the paper. References Ahmadi, G. (1976). Stability of a micro polar fluid layer heated from below. International Journal of Engineering Science, 14, 81–89. Alloui, Z., Vasseur, P., & Reggio, M. (2010). Natural convection of nanofluids in a shallow cavity heated from below. International Journal of Thermal Science, 50(3), 385–393. Buongiorno, J. (2006). Convective transport in nanofluids. ASME Journal of Heat Transfer, 128, 240–250. Chandrasekhar, S. (1961). Hydrodynamic and hydromagnetic stability. Oxford: Oxford University press. Choi, S. (1995). Enhancing thermal conductivity of fluids with nanoparticles. In Siginer, D.A., Wang, H.P., (Eds). Developments and applications of NonNewtonian flows, ASME FED 231/MD Vol. 66 (pp. 99–105). Dhananjay Agrawal, G. S., & Bhargava, R. (2011). Rayleigh–Bénard convection in nanofluid. International Journal of Applied Mathamatics and Mechanics, 7(2), 61–76. Kim, J., Kang, Y. T., & Choi, C. K. (2004). Analysis of convective instability and heat transfer characteristics of nanofluids. Physics of Fluid, 16, 2395–2401. Nield, D. A., & Kuznetsov, A. V. (2010). The onset of convection in a horizontal nanofluid layer of finite depth. European Journal of Mechanics B/Fluids, 29, 217–233. Qin, Y., & Kaloni, P. N. (1992). A thermal instability problem in a rotating micropolar fluid. International Journal of Engineering Science, 30, 1117–1126. Rajagopal, K. R., Saccomandi, G., & Vergori, L. (2009). On the Oberbeck- Boussinesq approximation in fluids with pressure-dependent viscosities. Nonlinear Analysis: Real World Applications, 10, 1139–1150. Tzou, D. Y. (2008a). Thermal instability of nanofluids in natural convection. International Journal of Heat and Mass Transfer, 51, 2967–2979. Tzou, D. Y. (2008b). Instability of nanofluids in natural convection. ASME Journal of Heat Transfer, 130, 372–401.